Let r be a system consisting of rational expressions. which operations are closed under r

We say that a set of numbers is closed under an operation when, no matter which two numbers we pick to use it on, we always end up with another number in the set.

The natural numbers, the positive whole numbers, are the first kind you ever learn about - 1, 2, 3, 4, 5, 6, etc. We say that the natural numbers are closed under addition because the sum of any two natural numbers is another natural number (e.g. 2 + 3 = 5, 67 + 52 = 109, etc.) We can also say that they're closed under multiplication, since the product of any two of them gives us another natural number (2 x 3 = 6, 10 x 7 = 70, 54 x 65 = 3510).

The natural numbers are not closed under subtraction, though, since problems like 3 - 5 = ? have no natural number solutions. We have to extend our number system to the integers, the set of all positive and negative whole numbers including 0, to make that solution possible, and we can then find that 3 - 5 = -2, an integer. We can now say that the integers are closed under addition, subtraction, and multiplication.

There are still situations we can't handle with just the integers, though. 3 ÷ 5 = ? has no solution in the integers since the solution isn't a whole number. To fix this, we extend our system again and finally reach the rational numbers, numbers written as ratios between integers (1/2, 3/6, -7/10, etc.) We can now add, subtract, multiply, and divide any two rational numbers and end up with another rational number, so we can say addition, subtraction, multiplication, and division are closed under R.

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